Anisotropic photon and electron scattering without ultrarelativistic approximation

Interactions between photons and electrons are ubiquitous in astrophysics. Photons can be downscattered (Compton scattering) or upscattered (inverse Compton scattering) by moving electrons. Inverse Compton scattering, in particular, is an essential process for the production of astrophysical gamma rays. Computations of inverse Compton emission typically adopts an isotropic or an ultrarelativistic assumption to simplify the calculation, which makes them unable to broadcast the formula to the whole phase space of source particles. In view of this, we develop a numerical scheme to compute the interactions between anisotropic photons and electrons without taking ultrarelativistic approximations. Compared to the ultrarelativistic limit, our exact results show major deviations when target photons are downscattered or when they possess energy comparable to source electrons. We also consider two test cases of high-energy inverse Compton emission to validate our results in the ultrarelativistic limit. In general, our formalism can be applied to cases of anisotropic electron-photon scattering in various energy regimes, and for computing the polarizations of the scattered photons.

Figure 1

A schematic diagram that depicts the IC scattering for the case of a point source of target photons. ${\overrightarrow{k}}_1$, ${\overrightarrow{k}}_2$, and $\overrightarrow{\beta }$ are the directions of target photon, scattered photon and source electron. See the text for a detailed description of the variables. Note that in general ${\overrightarrow{k}}_1$, ${\overrightarrow{k}}_2$, and $\overrightarrow{\beta }$ do not lie in the same plane, see Fig. 2.

Figure 2

A diagram for the IC scattering between a photon and an electron. With the assumptions made in Eq. (4), we can choose to align the target photon direction with $+z$-axis and set $x–z$ plane as the scattering plane.

Figure 3

The differential cross sections $E_2^2$ $d\sigma /d{E}_2$ against scattered photon energy $E_2$ in the Thomson regime $E_1\ll E_e$ (Sec. 3a1). Left: target photon energy $E_1={10}^{-6}\mathrm{MeV}$, source electron Lorentz factor $\gamma =1.5$. BG70 refers to the Eq. (2.48) in [30], the blue line marks the value of $E_1$. See paragraph for more details; Right: target photon energy $E_1={10}^{-6}\mathrm{MeV}$, source electron Lorentz factor $\gamma =100$.

Figure 4

Similar to Fig. 3, but for the KN regime, $E_1^{\prime }\gtrsim m_e$, $E_1\lesssim E_e$ (Sec. 3a2). Left: $E_1=0.5\mathrm{MeV}$ and $\gamma =1.5$, which corresponds to mildly-relativistic scattering; Right: $E_1=100\mathrm{MeV}$ and $\gamma =500$, which corresponds to ultrarelativistic scattering.

Figure 5

Similar to Fig. 3, but for the trans-Compton regime, $E_1>E_e$ (Sec. 3a3). Left: $E_1=1\mathrm{MeV}$, $\gamma =1.5$, corresponds to mildly-relativistic scattering; Right: $E_1=100\mathrm{MeV}$, $\gamma =100$, corresponds to ultrarelativistic scattering.

Figure 6

The LOS solar IC intensity at 0.3° away from the center of the sun. The red spectrum labeled with “stellarics” refers to the solid blue curve in [20] Fig. 4. The intensity spectrum in this work was computed using Eq. (1) with the same solar-photon and cosmic-electron spectrum in [58]. In particular, a thermal spectrum from ${10}^{-7}\mathrm{MeV}$ to ${10}^{-5}\mathrm{MeV}$ was used for the photon field. An electron spectrum from ${10}^1–{10}^5\mathrm{MeV}$ that follows the curve fitted with AMS02 data in Fig. 3 of [20], was used.

Figure 7

A decomposition of LOS solar IC spectrum. The decomposed spectra with $\gamma =1–1.5$, $\gamma =1.5–5$, $\gamma =5–10$, $\gamma =10–25$, and $\gamma =25–100$ are given by the magenta, blue, green, cyan, and yellow curves, respectively. Solid lines represent the stellarics results and our results (general formalism) are shown in dashed lines. Linear vertical scale is used to highlight the mildly-relativistic correction.

Figure 8

The SSC emission spectrum at observation angle 0.5° away from the jet with $E_{1,\mathrm{JF}}$ from ${10}^{-3}\mathrm{MeV}$ to ${10}^3\mathrm{MeV}$ and ${\gamma }_{\mathrm{JF}}$ from 1 to ${10}^4$. Both of target photons and electrons are assumed to be isotropic in JF. The emission is plotted in scattered photon energy in observer frame $E_2$. The red line gives the total emission. The individual contributions from ${\gamma }_{\mathrm{JF}}=1–{10}^1$, ${10}^1–{10}^2$, ${10}^2–{10}^3$, and ${10}^3–{10}^4$ are presented in the black, blue, cyan, and yellow lines, respectively. The transition points $P_1$ and $P_2$ for the blue line are highlighted. See the text for details about the decomposition spectrum and transition points.